Table Of ContentTAIL DECAY FOR THE DISTRIBUTION OF THE ENDPOINT OF A
DIRECTED POLYMER
THOMAS BOTHNER AND KARL LIECHTY
3 Abstract. We obtain an asymptotic expansion for the tails of the random variable =
01 argmaxu∈R(A2(u)−u2)whereA2istheAiry2process. UsingtheformulaofSchehr[37]Tthat
connects the density function of to the Hastings-McLeod solution of the second Painlev´e
2 equation, we prove that as t T , P( > t) = Ce−34ϕ(t)t−145/32(1+O(t−3/4)), where
n → ∞ |T|
ϕ(t)=t3 2t3/2+3t3/4, and the constant C is given explicitly.
a −
J
4
]
h 1. Introduction and statement of the main result
p
-
h Directedpolymers inarandommedium (DPRM)were introduced byHuse andHenley [27]
t to describe domain walls in a ferromagnetic Ising model with random impurities, sometimes
a
m called a dirty ferromagnet. In the two dimensional Ising model, a typical domain wall is a
[ lattice path in the plane, and DPRM in 1+1 dimensions is a statistical mechanical model
2 whose states are such paths. For concreteness, we consider the square lattice N N, and the
×
v graph with edges connecting nearest neighbors, so that the midpoints of the vertical edges
6 have the Cartesian coordinates (i,j +1/2) for some i,j N, and the horizontal edges have
1
8 the coordinates (i + 1/2,j) for some i,j N. On each∈vertex, we place a random weight
3 ε independently from some distribution,∈and consider some set of allowable lattice paths Γ
. ij
2 which originate at (0,0) and always move up and to the right. We then define the random
1
Gibbs measure on Γ as follows. For a path P Γ,
2
∈
1
:
v 1 1
µ(P) = exp ε ,
i ij
X Z T
(i,j) P
r X∈
a
where
1
Z = exp ε ,
ij
T
P Γ (i,j) P
X∈ X∈
is the (random) partition function, and T > 0 denotes temperature. For the set of allowable
paths Γ, usually we consider
Γ Γ = up-right paths ending at the point (m,n) ,
(m,n)
≡ { }
Date: January 8, 2013.
Part of this work was done during the workshop “Vector equilibrium problems and their applications
to random matrix models” hosted at the American Institute of Mathematics in April 2012. Both authors
would like to thank the organizers for their hospitality. KL would also like to thank Jinho Baik for helpful
comments.
1
2 THOMAS BOTHNER ANDKARLLIECHTY
in which case the model is said to have point-to-point geometry, or we consider
Γ Γ = up-right paths of length n ,
n
≡ { }
in which case the model is said to have point-to-line geometry.
In the limit as T 0, the Gibbs measure becomes a delta function on the path with the
→
greatest weight, and the randomness in the model comes entirely from the random weights
ε . This is known as directed last passage percolation. In the case of point-to-line directed
ij
last passage percolation with geometric or exponential weights on sites, it has been proven
[28] that the limiting fluctuations of both the energy of the maximizing path, and of the
location of the endpoint of the polymer can be described in terms of the Airy process.
2
The Airy process [34], which we denote (u), is a stationary process whose marginal
2 2
A
distributions are the Gaussian unitary ensemble (GUE) Tracy-Widom distribution [40], and
is expected to be a universal process governing the limiting spatial fluctuations of random
growth models in the Kardar-Parisi-Zhang (KPZ) universality class [29] in 1+1 dimension.
This has been proven in the case of the polynuclear growth model [28, 34]. Now let and
M
be the random variables defined by
T
:= max (u) u2 ,
2
M u R A −
∈
and (cid:0) (cid:1)
:= argmax (u) u2 .
2
T u R A −
∈
Then describesthelimitingfluctuationsof(cid:0)theenergyof(cid:1)themaximizingpathingeometric
M
point-to-line last passage percolation. The limiting fluctuations of the endpoint of the path
are described by . This fact was proved by Johansson [28] assuming that the maximum of
T
(u) u2 is attained at a unique point almost surely. This assumption was later proved by
2
A −
Corwin and Hammond [15]. For DPRM at finite temperature, there are some recent results.
For a continuum version anda semi-discrete version [33] ofDPRM, as well as discrete DPRM
with log-Gamma weights, the correct scaling exponents and a limit theorem for the free
energy have been obtained at finite temperature [6, 1, 11, 38, 10, 16, 39]. A limit theorem
for the fluctuations of the endpoint of the polymer with point-to-line geometry, however,
has only been proven in the case of geometric or exponential last passage percolation [28].
Nonetheless, isexpected to governthe fluctuations oftheend-point of apolymer inDPRM
T
at finite temperature as well, for a wide range of random weights ε . For a review of the
ij
KPZ universality class in the physical literature, see [25]. For a more recent review in the
mathematics literature, see [14].
Remark: Although has only been proven rigorously to govern endpoint fluctuations in
T
the case of geometric last passage percolation, a formula for the endpoint fluctuations of a
continuum version of DPRM was recently obtained nonrigorously by Dotsenko [20]. This
formula is equivalent to the known formulas for , given in (1.4) and (1.10) below. The
T
author of that paper was unable to prove this equivalence, and for the sake of completeness
in the literature, we give a short proof in Appendix C.
The distribution of is, up to rescaling by a constant, the same as the Tracy-Widom
M
distribution for the Gaussian orthogonal ensemble (GOE) [41]. Specifically, we have
P( t) = (22/3t),
1
M ≤ F
TAIL DECAY FOR THE DISTRIBUTION OF THE ENDPOINT OF A DIRECTED POLYMER 3
where is the Tracy-Widom GOE distribution function, defined below in (1.5) and (1.12).
1
F
This fact was proved by Johansson [28] by first proving a functional limit theorem for the
convergence of the polynuclear growth (PNG) model to the Airy process and using connec-
2
tions between PNG and the longest increasing subsequence of a random permutation found
by Baik and Rains [5]. A more direct, although nonrigorous, proof was given in [24] by
analyzing the fluctuations of nonintersecting Brownian excursions. A rigorous direct proof
based on the explicit determinantal formula for the Airy process was given in [17], and the
2
approach of [24] was made rigorous in [30]. Since the Tracy-Widom GOE distribution has
been well studied over the past 15-20 years, a lot is known about the distribution of . In
M
particular, the asymmetric tail behavior of is
1
F
e 2s3/2
1 −3 1+O s 3/2 , as s +
−
− 4√πs3/2 → ∞
(cid:18) (cid:19)
(s) = (1.1)
F1 s3 s3/2 (cid:0) (cid:1)
τ1e−|2|4 −|3|√2 1+O s 3/2 , as s ,
−
s 1/16 | | → −∞
| | (cid:18) (cid:19)
where (cid:0) (cid:1)
e12ζ′(−1)
τ = ,
1
211/48
and ζ( ) denotes the Riemann zeta-function. Similar formulas exist for the Tracy-Widom
·
GUE and Gaussian symplectic ensemble (GSE) distribution functions, see [2, 3, 18]. Re-
cently, some similar formulas have appeared for the general β Tracy-Widom distributions as
well [12, 13, 21].
Much less attention has been dedicated to the study of . Exact expressions for the
T
joint distribution of ( , ) were obtained in two recent papers: in [32] by Moreno Flores,
M T
Quastel, and Remenik; and in [37] by Schehr. The formula of [32] involves the Airy function
andtheresolvent ofanassociatedoperator andis derived rigorously, while theformula of[37]
involves a solution to the Lax pair for the Painlev´e II equation, and is derived nonrigorously.
It was shown in [4] that these formulas are indeed the same, and therefore the formula of
[37] is put on rigorous footing.
Let us describe the two formulas. In order to do so, we first need to fix some notation
which we will use throughout the paper. Let Pˆ(m,t) denote the joint density function of
( , ). Let Ai(x) be the Airy function [7], and let B be the integral operator acting on
s
M T
L2[0, ) with kernel
∞
B (x,y) = Ai(x+y +s), (1.2)
s
where s R is a parameter. It is known that 1 B is invertible for any s R. Let
s
∈ − ∈
ρ (x,y) = (1 B ) 1(x,y), x,y 0. (1.3)
s s −
− ≥
Define, for t,m R,
∈
ψ(x;t,m) = 2ext[tAi(t2 +m+x)+Ai(t2 +m+x)].
′
Then the formula of [32] is
Pˆ(m,t) = 21/3 (22/3m) ∞ ∞ψ(21/3x ; t,m)ρ (x ,x )ψ(21/3x ;t,m)dx dx . (1.4)
F1 1 − 22/3m 1 2 2 1 2
Z0 Z0
4 THOMAS BOTHNER ANDKARLLIECHTY
In terms of the operator B , the GOE Tracy-Widom distribution function equals
s 1
F
(s) = det(1 B ), (1.5)
1 s
F −
see [22, 36].
We now present the formula of [37]. To this end let q(s) be the particular solution of the
second Painlev´e equation,
q (s) = sq(s)+2q(s)3,
′′
satisfying
q(s) Ai(s), as s + .
∼ → ∞
ThisparticularsolutionisknownastheHastings-McLeodsolution[26,23],anditsuniqueness
and global existence are well established. Consider now the following Lax-system (cf. [8])
associated to the Hastings-McLeod solution of the Painlev´e II equation, i.e. the system of
linear differential equations for a two-dimensional vector Φ = Φ(ζ,s),
∂Φ ∂Φ
= AΦ , = BΦ , (1.6)
∂ζ ∂s
where the 2 2 matrices A = A(ζ,s) and B = B(ζ,s) are given by
×
4ζq 4ζ2 +s+2q2 +2q
A(ζ,s) = ′ , (1.7)
4ζ2 s 2q2 +2q 4ζq
′
(cid:18) − − − − (cid:19)
and
q ζ
B(ζ,s) = . (1.8)
ζ q
(cid:18) − − (cid:19)
The above system (1.6) is overdetermined, and the compatibility of the equations implies
that q(s) solves the Painlev´e II equation. Now let Φ = Φ1 be the unique solution of (1.6)
Φ2
which satisfies the real asymptotics
(cid:0) (cid:1)
4 4
Φ (ζ;s) = cos ζ3 +sζ +O(ζ 1), Φ (ζ;s) = sin ζ3 +sζ +O(ζ 1),
1 − 2 −
3 − 3
(cid:18) (cid:19) (cid:18) (cid:19)
as ζ for s R. Such a solution exists (see e.g. [8, 19, 23]), and it further satisfies
→ ±∞ ∈
the property that Φ (ζ;s) and Φ (ζ;s) are real for real ζ and s, as well as the symmetry
1 2
relations
Φ ( ζ;s) = Φ (ζ;s), Φ ( ζ;s) = Φ (ζ;s), ζ C,s R.
1 1 2 2
− − − ∈ ∈
Define for non-negative w and real s,
h(s,w) := ∞ζΦ (ζ;s)e wζ2dζ. (1.9)
2 −
Z0
The formula of [37] for the joint density of and is
M T
Pˆ(m,t) = 4P(22/3m,24/3t) , (1.10)
where
4
∞
P(s,w) = (s) h(u,w)h(u, w)du, (1.11)
π2F1 −
Zs
TAIL DECAY FOR THE DISTRIBUTION OF THE ENDPOINT OF A DIRECTED POLYMER 5
and h(s, w), w > 0 is understood as the analytic continuation of h( ,w) to the negative real
− ·
axis. In terms of the Painlev´e function q(s), the Tracy-Widom GOE distribution function
can be written as [41]
1
F
1 1
(s) = exp ∞q(x)dx ∞ ∞q(x)2dxdt . (1.12)
1
F −2 − 2
(cid:20) Zs Zs Zt (cid:21)
The marginal density function for equals
T
Pˆ(t) := ∞ Pˆ(m,t)dm,
Z
−∞
and using (1.10), we have
Pˆ(t) := 24/3P(24/3t), (1.13)
where we introduced
∞
P(w) = P(s,w)ds. (1.14)
Z
−∞
ˆ
Our main result in the present paper is an asymptotic expansion of the density function P(t)
for large t.
Theorem 1.1. As t , the marginal density function Pˆ(t) satisfies
→ ∞
15
Pˆ(t) = τe−34ϕ(t)t−81/32 1+ +O(t−3/2) ,
4t3/4
(cid:18) (cid:19)
which extends to a full asymptotic series in powers of t 3/4. The function ϕ(t) is
−
ϕ(t) = t3 2t3/2 +3t3/4,
−
and the constant τ is given by
τ = 2−29/6e5/4e21ζ′(−1)π3/2,
with ζ(z) denoting the Riemann zeta-function.
Integrating the above density, Theorem 1.1 implies the following corollary.
Corollary 1.2. As t ,
→ ∞
15
P( > t) = Ce−43ϕ(t)t−145/32 1+ +O(t−3/2) . (1.15)
|T | 4t3/4
(cid:18) (cid:19)
where C = τ/2.
Remark: The leading decay order of e ct3 in (1.15) was first predicted in the physics
−
literature by Halpin-Healy and Zhang [25], compare also [31]. The first rigorousconfirmation
of this rate of decay appeared in the paper [15] of Corwin and Hammond, in which they
give e ct3 as an upper bound on P( > t), although they do not give the value of the
−
|T|
constant c. In the paper [37] of Schehr, the author uses the formula (1.10) to find that the
leading coefficient is c = 4/3, although he did not employ complete and rigorous estimates.
Recently Quastel and Remenik [35] rigorously obtained explicit bounds on c, and remarked
that they believe that the correct rate of decay is in fact 4/3. Corollary 1.2 confirms this
rigorously, and in addition gives subleading terms and constants. In principle, all terms in
6 THOMAS BOTHNER ANDKARLLIECHTY
the asymptotic expansion are computable by the methods of this paper, but calculations
become more involved.
The setup for the remainder of the paper is a follows. We will prove Theorem 1.1 using
(1.11), (1.13) and (1.14). In Section 2 we evaluate h(s,w) as w by analyzing the
→ ∞
system (1.6) close to the origin. In Section 3 we prepare h(s, w) for asymptotic analysis
−
as w using the global identity from [4] which allows us to express Φ (ζ;s) in terms of
2
→ ∞
the resolvent of a Hankel operator on L2[0, ) whose kernel is constructed out of the Airy
∞
function. Then in Section 4 we split P(w) into two parts, one which can be estimated using
asymptotics (1.1) of the Tracy-Widom distribution function (s), and another which we
1
F
evaluate asymptotically by Laplace’s method. Theorem 1.1 then follows via (1.13), and the
proof of Corollary 1.2 will be given in Section 5.
2. Expansion of h(s,w) as w +
→ ∞
With the change of variables λ = ζ√2w, (1.9) becomes
1 ∞ λ λ2
h(s,w) = λΦ2 ;s e− 2 dλ. (2.1)
2w √2w
Z0 (cid:18) (cid:19)
We thus see that we need the Taylor expansion of Φ (ζ;s) at ζ = 0. To this end let us
2
consider (1.6) as a system for a 2 2 matrix-valued function Ψ,
×
2 1
∂Ψ ∂Ψ
= A (s)ζn Ψ, = B (s)ζn Ψ, Ψ(ζ,s) = Φ(ζ,s),Φˆ(ζ,s) , (2.2)
n n
∂ζ ∂ζ
(cid:18)n=0 (cid:19) (cid:18)n=0 (cid:19)
X X (cid:0) (cid:1)
where
0 v(s)+4q (s) 4q(s) 0 0 4
A = ′ , A = , A = ,
0 v(s) 0 1 0 4q(s) 2 4 0
(cid:18)− (cid:19) (cid:18) − (cid:19) (cid:18)− (cid:19)
q(s) 0 0 1
B = , B = ,
0 0 q(s) 1 1 0
(cid:18) − (cid:19) (cid:18)− (cid:19)
are determined from (1.7) and (1.8), and
v(s) := s+2q(s)2 2q (s). (2.3)
′
−
The matrix function Ψ can alternatively be defined as the solution of a certain oscillatory
Riemann-Hilbert problem, see Appendix A. Such Riemann-Hilbert formulation appears in
one form or another throughout the literature on the integrable structure of Painlev´e II, and
is the basis for many results (cf. [23]). In particular, we will use the facts that
Ψ( ζ;s) = σ Ψ(ζ;s)σ , ζ C, s R, (2.4)
3 3
− ∈ ∈
and
Ψ(0;s) = e−σ3 s∞q(t)dt, (2.5)
R
which are justified in Appendix A.
Since the coefficient functionA(ζ,s)is analytic at theorigin, there exists a solutionΨ(ζ,s)
of equation (2.2) which is holomorphic in some neighborhood of the origin and uniquely
TAIL DECAY FOR THE DISTRIBUTION OF THE ENDPOINT OF A DIRECTED POLYMER 7
determined by its value Ψ(0,s) = Ψ (s) (cf. [42]). The Taylor expansion of Ψ about ζ = 0
0
is
∞
Ψ(ζ,s) = Ψ (s)ζn, ζ < ρ, ρ > 0. (2.6)
n
| |
n=0
X
Inserting (2.6) into (2.2) and comparing coefficients yields for n Z first
0
∈ ≥
(n+1)Ψ = A Ψ (s)+A Ψ (s)+A Ψ (s), Ψ (s) = Ψ (s) 0, (2.7)
n+1 0 n 1 n 1 2 n 2 1 2
− − − − ≡
and secondly
dΨ 1 0 0 i
n
= q(s)σ Ψ (s)+iσ Ψ (s), σ = , σ = − .
ds 3 n 2 n−1 3 0 1 2 i 0
(cid:18) − (cid:19) (cid:18) (cid:19)
The latter differential recursion implies
Ψ0(s) = e−σ3 s∞q(x)dxC0,
R
for some 2 2 invertible matrix valued constant C . In fact, from (2.5) we see that C = I.
0 0
×
In order to obtain the rest of the Taylor coefficients, we use the recursion (2.7) to write Ψ
k
explicitly in terms of the matrices A ,A ,A . The first two coefficients are
0 1 2
1
Ψ1(s) = A0e−σ3Rs∞q(x)dx, Ψ2(s) = 2 A20 +A1 e−σ3Rs∞q(x)dx. (2.8)
At this point we would like to connect the coefficients(cid:0)Ψ (s) to(cid:1)the required values of Φ (ζ;s)
n 1
and Φ (ζ;s). The aformentioned global symmetry relation (2.4) implies
2
Ψ (s) = ( 1)nσ Ψ (s)σ , n Z ,
n 3 n 3 0
− ∈ ≥
i.e. all coefficients Ψ (s) are diagonal whereas all coefficients Ψ (s) are off-diagonal. We
2n 2n+1
therefore have
Φ (ζ;s) = ∞ Ψ(11)(s)ζ2n, Φ (ζ;s) = ∞ Ψ(21) (s)ζ2n+1, Ψ = Ψ(ij) 2 , ζ < ρ.
1 2n 2 2n+1 n n i,j=1 | |
n=0 n=0
X X (cid:0) (cid:1) (2.9)
From (2.7) (see explicitly (2.8)),
Ψ(22n1+)1(s) = Pn s,q(s),q′(s) e−Rs∞q(x)dx, (2.10)
with polynomials P in three variables. Integrating (2.1), we obtain the following Proposi-
n (cid:0) (cid:1)
tion.
Proposition 2.1. As w + , the function h(s,w) satisfies the asymptotic expansion
→ ∞
√π ∞
h(s,w) = 4w3/2e−Rs∞q(x)dx Qn(s,q(s),q′(s))w−n, (2.11)
n=0
X
where
(2n+1)!
Qn(s,q(s),q′(s)) = eRs∞q(x)dxΨ(22n1+)1(s) n!4n ,
is a polynomial in 3 variables. This expansion is uniform in s on compact subsets of the real
line. The first two terms in the expansion (2.11) are
v(s)3
Q (s,q(s),q (s)) = v(s), Q (s,q(s),q (s)) = +v(s) v(s)q (s)+q(s) 2, (2.12)
0 ′ 1 ′ ′
− 4 −
(cid:0) (cid:1)
8 THOMAS BOTHNER ANDKARLLIECHTY
where v(s) is defined in (2.3). Moreover, as s , the polynomial Q (s,q(s),q (s))
n ′
→ −∞
satisfies
Q (s,q(s),q (s)) = O ( s)(n 1)/2 . (2.13)
n ′ −
−
Proof. Replacing Φ in (2.1) by its power series (2(cid:0).9) gives (cid:1)
2
(21)
h(s,w) = 1 ∞ Ψ2n+1(s) ∞λ2n+2e−λ22 dλ
(2w)3/2 (2w)n
n=0 Z0
X
(21)
√π ∞ Ψ (s)(2n+1)!
= 2n+1 w n.
−
4w3/2 4nn!
n=0
X
(21)
Using the structure (2.10) of Ψ (s), we can thus write, as w ,
2n+1 → ∞
√π ∞
h(s,w) = 8w3/2e−Rs∞q(x)dx Qn(s,q(s),q′(s))w−n.
n=0
X
For the asymptotics (2.13), we use [19] . Namely as s ,
†
→ −∞
s 1 73 10657
q(s) = − 1+ + +O(s 12) ,
−
2 8s3 − 128s6 1024s9
r (cid:18) (cid:19) (2.14)
1 5 803 181169
q (s) = 1 + +O(s 12) ,
′ −
−23/2√ s − 8s3 128s6 − 1024s9
− (cid:18) (cid:19)
from which it follows
1 ( s) 3/2 5 9( s) 9/2 803 1323( s) 15/2
− − −
v(s) = 1 − − + −
√ 2s − 2√2 − 8s3 − 4√2 128s6 − 32√2
− (cid:18)
181169
+O ( s) 21/2 .
−
−1024s9 −
(cid:19)
(cid:0) (cid:1)
From these we find that, as s ,
→ −∞
O(( s)n/2) 0
Ψ2n(s) = −0 O(( s)n/2) e−σ3Rs∞q(x)dx,
(cid:18) − (cid:19)
0 O(( s)(n 1)/2)
Ψ2n+1(s) = O(( s)(n 1)/2) − 0 − e−σ3Rs∞q(x)dx,
−
(cid:18) − (cid:19)
(cid:3)
which is easily checked by induction using the recursion (2.7).
Let us now consider h(s,w) as w .
→ −∞
We would like to point out the following small detail. In [2] the coefficient of O(s−9) is given as 10219,
† 1024
whereas in [19] and [40] it is 10657.
1024
TAIL DECAY FOR THE DISTRIBUTION OF THE ENDPOINT OF A DIRECTED POLYMER 9
3. Airy formula for h(s,w) and discussion
In order to analyze h(s,w) for large negative values of w it is convenient to use a formula
for Φ which was obtained in [4]. Let us review the notation of [4]. As in (1.2), let B be
2 s
the integral operator acting on L2[0, ) with kernel
∞
B (x,y) = Ai(x+y +s).
s
Let A := B2 be the Airy operator acting on L2[0, ), which has kernel
s s ∞
∞
A (x,y) = Ai(x+s+ξ)Ai(y +s+ξ)dξ
s
Z0
Ai(x+s)Ai(y +s) Ai(x+s)Ai(y +s)
′ ′
= − .
x y
−
Define the functions Q and R as
Q := (1 A ) 1B δ , R := (1 A ) 1A δ , (3.1)
s − s 0 s − s 0
− −
where δ is the Dirac delta function at zero. As in [4] we use the convention that the Dirac
0
delta function satisfies δ (x)f(x)dx = f(0) for functions f which are right-continuous
[0, ) 0
at 0. Introduce also the fu∞nction
R
4
Θ(x) := sin ζ3 +(s+2x)ζ .
− 3
(cid:18) (cid:19)
Proposition 2.1 of [4] is
Φ (ζ,s) = Θ(0)+ Θ,R+Q , (3.2)
2
h i
where the functions Q and R are defined in (3.1), and , is the inner product on L2[0, ).
h· ·i ∞
Notice that the function R+Q can be written in terms of the resolvent kernel of the operator
B . Indeed, let
s
R := (1 B ) 1 1,
s s −
− −
be the resolvent of the operator B . It is an integral operator as well, and let us use K to
s s
denote the function R δ . We then have
s 0
R+Q = (1 B2) 1(B +B2)δ = (1 B ) 1B δ = (1 B ) 1(1 (1 B ))δ
− s − s s 0 − s − s 0 − s − − − s 0 (3.3)
= ((1 B ) 1 1)δ = K ,
s − 0 s
− −
and (3.2) becomes
Φ (ζ,s) = Θ(0)+ Θ,K . (3.4)
2 s
h i
Substituting (3.4) into (1.9) gives
4
h(s,w) = ∞ζsin ζ3 +sζ e wζ2dζ + ∞ζ Θ,K e wζ2dζ
− s −
− 3 h i
Z0 (cid:18) (cid:19) Z0 (3.5)
= a(s,w)+ ∞a(s+2x,w) (1 B ) 1T Ai (x)dx,
s − s
−
Z0
(cid:16) (cid:17)
with
4
a(y,w) = ∞ζsin ζ3+yζ e wζ2dζ, y R, (3.6)
−
− 3 ∈
Z0 (cid:18) (cid:19)
and the translation operator
T f (x) = f(x+s).
s
(cid:0) (cid:1)
10 THOMAS BOTHNER ANDKARLLIECHTY
When w lies off of the non-negative real axis (3.6) diverges. In order to obtain the analytical
continuation of a( ,w) to the complex w-plane, set ζ = iz. Then
· −
1 i 4 1 i
a(y,w) = ·∞zsinh z3 yz ewz2dz = ·∞ ze43z3+wz2−yzdz,
i 3 − 2i
Z0 (cid:18) (cid:19) Z i
−·∞
and by Cauchy’s theorem we can deform the contour of integration such that
1
a(y,w) = ze43z3+wz2−yzdz, (3.7)
2i
Z
L
where is any contour that starts at the point infinity with argument arg z = π (see
L −3
Figure 1) and ends at the conjugated point. We make the following observations. First (3.7)
L
Figure 1. The integration contour chosen in (3.7). The dashed lines are
L
the rays arg z = π and arg z = π.
±6 ±3
converges absolutely and uniformly in any compact w domain, hence (3.7) determines the
analytic continuation of a( ,w) to the entire real line and it allows us to consider (3.5) now
for all w R. Secondly usi·ng the definition
∈
1
Ai(z) = e31t3−tzdt, z C,
2πi ∈
Z
L
we obtain (see [4] for a similar computation),
a(y,w) = π ew243+w4y Ai′ 2−2/3y +2−8/3w2 + w Ai 2−2/3y +2−8/3w2 , w,y R.
24/3 24/3 ∈
(cid:18) (cid:19)
(cid:0) (cid:1) (cid:0) (cid:1)
In particular, notice that, for y w2/4 we have, as w ,
≫ − → ∞
a(y, w) = √π√we−w243−w4y ∞ 4y n 3 m 2−2/3y +2−8/3w2 −32m
− − 8 w2 − 2
nX,m=0(cid:18) (cid:19) (cid:18) (cid:19) (cid:16) (cid:17)
1/4 1/4 2 3/2
d + − c exp 2 2/3y +2 8/3w2 , (3.8)
m m − −
× n n − 3
" #
(cid:18) (cid:19) (cid:18) (cid:19)
h i (cid:16) (cid:17)
